How to add weights to one specific input feature to ensure fair training in the network?
I think your approach to tackle this as an imbalanced problem is correct. The easiest thing you could do is to add weights to the samples, during training, so that the model "pays more attention" to the under-represented class. 

There are also a couple of other ways for this to be done: oversampling and undersampling, but initially, I'd focus on adding weights, since its easier to implement.